02-07-gradients

#eth/cil/theory

Gradients

Vector / Matrix notation

\nabla_{u} l = R v, \nabla_{v} l = R^{T} u, R := (u v^{T} - A)

\nabla^{2} l (u, v) = (\begin{matrix} | | v | |^{2} 1_{n \times n} & 2 u v^{T} - A \\ (2 u v^{T} - A)^{T} & | | u | |^{2} 1_{m \times m} \end{matrix})

At the origin:

\nabla^{2} l (0, 0) = (\begin{matrix} 0 & - A \\ - A^{T} & 0 \end{matrix})

This is scalar to the mulitdimensional case. The hessian at zero is not positive-semi-definit , unless $A$ is nilpotent (has all zero eigenvalues).

This problem is non-convex for all dimensions $m, n \geq 1$